ℓ-Adic properties of the partition function

Abstract

Ramanujanʼs famous partition congruences modulo powers of 5, 7, and 11 imply that certain sequences of partition generating functions tend ℓ-adically to 0. Although these congruences have inspired research in many directions, little is known about the ℓ-adic behavior of these sequences for primes ℓ ⩾ 13 . Using the classical theory of “modular forms mod p”, as developed by Serre in the 1970s, we show that these sequences are governed by “fractal” behavior. Modulo any power of a prime ℓ ⩾ 5 , these sequences of generating functions ℓ-adically converge to linear combinations of at most ⌊ ℓ − 1 12 ⌋ − ⌊ ℓ 2 − 1 24 ℓ ⌋ many special q-series. For ℓ ∈ { 5 , 7 , 11 } we have ⌊ ℓ − 1 12 ⌋ − ⌊ ℓ 2 − 1 24 ℓ ⌋ = 0 , thereby giving a conceptual explanation of Ramanujanʼs congruences. We use the general result to reveal the theory of “multiplicative partition congruences” that Atkin anticipated in the 1960s. His results and observations are examples of systematic infinite families of congruences which exist for all powers of primes 13 ⩽ ℓ ⩽ 31 since ⌊ ℓ − 1 12 ⌋ − ⌊ ℓ 2 − 1 24 ℓ ⌋ = 1 . Answering questions of Mazur, in Appendix A we give a new general theorem which fits these results within the framework of overconvergent half-integral weight p-adic modular forms. This result, which is based on recent works by N. Ramsey, is due to Frank Calegari.